bessel functions

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1、Chapter2BesselFunctions(1)(2)2.1Bessel,Neumann,andHankelFunctions:Jn(x),Nn(x),Hn(x),Hn(x)Besselfunctionsaresolutionsofthefollowingdi®erentialequation:x2y00+xy0+(x2¡º2)y=0(2.1)whichiscalledtheBessel'sdi®erentialequation.Thisisasecondorderdi®erentialequationandhastwolinearlyindepend

2、entsolutions.Anytwoofthefollowingfunctionsarelinearlyindependentsolutionsof(2.1)J(x)N(x)H(1)(x)H(2)(x)(2.2)ººººThus,thegeneralsolutionfor(2.1)canbewrittenasalinearcombinationofanytwooftheabovefunctions.Usuallythegeneralsolutioniswrittenasoneoftheseforms:AJ(x)+BN(x)AH(1)(x)+BH(2)(x

3、)(2.3)ººººWhenºisnotaninteger(º6=n)JºandJ¡ºarealsolinearlyindependentsolutionsof(2.1),however,weusuallyneveruseJ¡º.TheNeumannfunctionisrelatedtoJºandJ¡º:cosº¼Jº(x)¡J¡º(x)Nº(x)=(2.4)sinº¼cosº¼Jº(x)¡J¡º(x)Nn(x)=lim(2.5)º!nsinº¼andHankelfunctionsofthe¯rstandsecondkindarerelatedtoBess

4、elandNeumannfunctions:H(1)(x)=J(x)+jN(x)(2.6)ºººH(2)(x)=J(x)¡jN(x)(2.7)ºººWithavariabletransformationx=·½equation(2.1)canbetransformedinto:½2y00+½y0+(·2½2¡º2)y=0(2.8)whoseindependentsolutionsareJº(·½)andNº(·½).Whenº=nisanintegerJnandJ¡narenotindependentanymoreandwehave:J¡n(x)=(¡1)

5、nJn(x)N¡n(x)=(¡1)nNn(x)(2.9)Jn(¡x)=(¡1)nJn(x)Nn(¡x)=(¡1)n[2jJn(x)+Nn(x)]Plotsofthe¯rstthreeBesselfunctionsareshowninFig.2.1andthe¯rstthreeNeumannfunctionsareshowninFig.2.28AmirBorjiSpecialFunctionsandLaplaceEquation910.8J0(x)0.6J1(x)0.4J2(x)0.20−0.2−0.4−0.602468101214Figure2.1:Bes

6、selfunctionsofthe¯rstkind2.1.1SmallandLargeArgumentApproximationsSmallArgumentLimitjxj¿11³x´nJn(x)¼(2.10)n!2µ¶n(n¡1)!2Nn(x)¼¡n6=0(2.11)¼x2°xY0(x)¼ln°=1:78107241799:::Euler'sconstant(2.12)¼2LargeArgumentLimitjxjÀ1r³´2¼n¼Jn(x)¼cosx¡¡(2.13)¼x42r³´2¼n¼Nn(x)¼sinx¡¡(2.14)¼x422.1.2Orthog

7、onalityRelationshipsandFourier-BesselExpansionsBesselequation(2.8)canbewritteninthefollowingform:µ¶n2(½y0)0+·2½¡y=0(2.15)½n2Ifyoucompare(2.15)with(1.5),youcanseethatitisaSturm-Liouvilleequationwithp(½)=½,q(½)=¡,½w(½)=½,and¸=·2.Withappropriateboundaryconditionsona¯niteintervalsucha

8、s½2[a;b]wecanhaveaSturm-Liouville